Study of X-ray topography using the super-Borrmann effect

X-ray topography exerting the super-Borrnann effect is realized using synchrotron radiation and a high-resolution CMOS camera.


Introduction
The effect of anomalous transmission can be enhanced (Borrmann & Hartwig, 1965) if the Bragg condition is satisfied for the 111 and 1 111 reflections simultaneously in the wideangle diagram of perfect germanium (Ge) crystals with thickness t = 0.8 mm and 1.2 mm with Cu K radiation. Enhanced intensity spots for the 111 and 1 111 reflections appear at the Kossel line intersection of the T 111 and T 1 111 traces of the reflected beams, respectively. Furthermore, enhanced intensity spots for the 111 and 1 111 reflections appear on the R 111 and R 1 111 traces of the transmitted (refracted in the strict sense) beams, respectively, and are symmetrical to the intersection point of the T 111 and T 1 111 traces with respect to the respective reflecting planes. Calculation results and interpretation for the decrease in the absorption coefficient was provided (Hildebrandt, 1966(Hildebrandt, , 1967 for the enhanced spots in the 111; 1 111=200 three-beam case (/hkl means g-h when g = 111 and h = 1 111). Later, the theoretical understanding based on detailed calculation was advanced (Feldman & Post, 1972), and was confirmed experimentally (Uebach & Hildebrandt, 1969;Hildebrandt, 1978).
In terms of the application of the super-Borrmann effect to X-ray topography to image lattice defects in crystals, such as dislocations, as well as conventional X-ray topography, few reports have been published probably owing to a too low X-ray source intensity and large X-ray beam divergence to develop clear defect images for a wide visual field. However, since the availability of synchrotron radiation, X-ray topography can be applied for imaging lattice defects in crystals by choosing the correct X-ray wavelength. In addition, it was previously reported that topographs combined with a highspeed and high-resolution CMOS camera taken by employing forward-transmitted X-rays under multiple diffraction conditions (bright-field X-ray topographs) can reveal dislocations without noticeable image deformations (Tsusaka et al., 2016(Tsusaka et al., , 2019. Furthermore, as a major advantage, it is expected that the forward-transmitted X-rays riding on the super-Borrmann effect also reveal dislocations existing in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes such as (111) and ð 1 111Þ where conventional Lang X-ray topography is difficult to apply. Therefore, this study deals with X-ray topography performed under three-beam multiplediffraction conditions for thick Ge crystals using synchrotron radiation.

111;
1 111=200 three-beam case Fig. 1 shows an example of 111; 1 111=200 three-beam multiple diffraction in reciprocal space of a perfect Ge crystal. Note that the 1 111 beam is not drawn here considering that it passes symmetrical to the 111 beam with respective to the (100) plane of symmetry in order to simplify calculation of the absorption decreases controlling the super-Borrmann effect. Figs. 1(a) and 1(b) demonstrate two cases of different energy of the incident X-rays for E = E 1 and E = E 2 , respectively, which is higher than E 1 . The black dashed triangle in Fig. 1 comprises the original K o , K 111 and g 111 , where K o is the incident X-ray wavevector, K 111 is the 111-reflected X-ray wavevector and g 111 = K 111 À K o is the diffraction vector of the 111 reflection lying on the same plane. Rectangles OPQR and PP 0 O 0 O represent projections on ð01 1 1Þ and (100), respectively. The lengths of sides PQ, OP and PP 0 are given as follows, where ! is an elevation angle of K o (or K 111 ) from the rectangle OPQR parallel to the 0 1 11 À Á entrance surface. It is clear that both PQ and OP are independent of E; however, PP 0 becomes larger when E increases, as observed in Figs. 1(a) and 1(b). Then, we put a unit vector of K o as s o and unit vectors of the polarization components of K o as r o and p o , for horizontal and vertical polarizations, respectively. r o lies in the 01 1 1 À Á base plane and is perpendicular to s o . Therefore, p o is also perpendicular to s o and r o .
Next, we put K s 111½0 , K 111½0 and K 111½0 in Fig. 1 as components of K 111 in the s o , r o and p o directions, respectively. The magnitudes of these vectors were calculated as k cos 2, ð2= ffiffi ffi 3 p Þ k sin and 2k sin ½ð2=3Þ À sin 2 1=2 , respectively. The lengths of sides OX and XP were found to have the following values, Schematic of 111; 1 111=200 three-beam multiple diffraction in reciprocal space of Ge, where only the 111 reflection is represented, considering the 1 111 beam travels in a symmetrical direction with the plane of symmetry (100) at incident X-ray energies of E = E 1 in (a) and E = E 2 (> E 1 ) in (b) with rectangles OPQR and PP 0 O 0 O representing projections on the 01 1 1 À Á and (100) planes, respectively. 01 1 1 À Á : entrance surface; K o : incident X-ray wavevector; K 111 : 111-reflected X-ray wavevector; g 111 : diffraction vector of the 111 reflection.
Since K 1 111 is symmetrical with K 111 , jK 1 111½0 j = ÀjK 111½0 j; however, the s o , p o components of K 1 111 and K 111 are identical. Consequently, the refracted beam K o and two reflected beams K 111 and K 1 111 are summarized as follows,

Absorption coefficients in the three-beam diffraction cases
Since the calculation process for the absorption coefficient in the three-beam case under Cu K 1 radiation has already been provided by Authier (2001), explanation of the calculation will be kept to a minimum. Based on the fundamental equations of X-ray dynamical theory, the projections of the electric displacements D in the three-beam case to the plane normal to K o can be written for the three beams as follows, It is possible to rewrite the above equations by using excitation errors o , 111 , 1 111 , Since the 200 reflection and 2 200 reflectin are forbidden ( 200 = 2 200 = 0), the above equations are expressed as follows, The two relations are obtained from the second and third lines of equation (3) shown above, C in a similar way to that described by Authier (2001). By substituting D 111[o] and D 1 111 o ½ thus obtained in the first line of equation (3), we obtain a relation involving only D o . If we decompose D o into two components, D o , parallel to the plane of symmetry, and D o , perpendicular to the plane of symmetry, The first line of equation (2), which is can be replaced using two scalar values D o and D o , given as follows, Considering X can be separated into r o and p o , Therefore, we can derive the determinant as follows, and hence, inevitably, Note that the components of K o , K 111 and K 1 111 are given by equation (1) for r o polarization and for p o polarization.
In the two-beam case, When | o | = | 111 |, satisfying the Bragg condition exactly, For a cubic crystal such as Ge, the structure factor F 111 and Fourier component of the dielectric susceptibility 111 for the 111 reflection are given as follows, where f Ge is the atomic scattering factor of Ge, r e is the classical electron radius, is the wavelength of the X-rays, and V c is the volume of a unit cell of Ge. F 111_ r and F 111_ i are the real and imaginary parts of the complex number F 111 , respectively. The magnitudes of F 111 and 111 can be derived from the corresponding atomic scattering factors as follows, The minimum value of the absorption coefficient in the g, h=ðg À hÞ = 111; 1 111=200 three-beam case is given as follows, where o is the normal absorption coefficient and 111_i and o_i are imaginary parts of 111 and o , respectively. The minimum attenuation coefficient is A P min = exp½À e ðt= o Þ, where t is the slab thickness and o = n hkl Á s o is a direction cosine of the incident X-ray wavevector K o (its unit vector is s o ) to n hkl , the normal to the X-ray entrance surface. In the 111; 1 111=200 present three-beam case, it is found from Fig. 2 that the direction cosine o is expressed as for K o to n 0 1 11 . However, the X-ray energy in the present experimental case using synchrotron radiation was E = 10 keV and the Ge slab thickness was t = 0.05 cm with the (100) entrance surface. Because the lattice parameter of Ge is a = 0.56754 nm and the Bragg angle of the 111 reflection becomes 111 = 7.2458 leading to sin 111 = 1:23984=½2 a= ffiffi ffi 3 p À Á E = 0.12613, we can derive o for n hkl = n 001 from Fig. 2 as follows, which corresponds to 31.608 as an angle between n 001 and s o . In this case, the polarization factors P for the r o and p o components are introduced from equation (6) as for r o polarization and for p o polarization This makes it possible to calculate the effective absorption coefficient e and the minimum attenuation coefficient A P min in the 111; 1 111=200 three-beam case for the Ge slab having the (001) entrance surface with thickness of 0.05 cm, as demonstrated in Table 1 Schematic of the relation between K o , the wavevector of the incident X-rays, and n 001 , the (001) surface normal, or n 0 1 11 , the 0 1 11 À Á surface normal. Angles ð=2Þ À ! and ! À ð=4Þ correspond to direction cosines o of K o to n 001 and n 0 1 11 , respectively. atten2.html). Furthermore, it was observed that A P min for both polarizations in the three-beam case are approximately 20 times the values in the two-beam case, due to which the phenomenon is called the super-Borrmann effect.

Topography experiment using the super-Borrmann effect for Ge crystals
To take X-ray topographs with minimized image deformation, we employed forward-transmitted (but refracted) X-rays that satisfied the Bragg conditions for the two {111} adjacent planes lying symmetrically with respect to the {100} plane of symmetry, as demonstrated in Fig. 3. An X-ray diffraction goniometer with an X-ray source of approximately 1.2 mm Â 1.2 mm was used with 10 keV X-rays from the synchrotron radiation through a silicon double-crystal monochromator at the BL24XU8 beamline of SPring-8 (Tsusaka et al., 2001), similar to previously reported multiple-beam diffraction topography (Tsusaka et al., 2016(Tsusaka et al., , 2019. In order to avoid the harmonics of the incident synchrotron beam, the usual detuning treatment was carried out before carrying out the topography experiment. Various interference patterns on diffracted and transmitted images with defect appearance were also studied using a coherent X-ray beam under multiple-diffraction conditions Okitsu, 2003). However, in the present case, topographic images were taken directly by the forwardtransmitted X-ray beam instead of the diffracted X-ray beam using an X-ray imaging detector (Hondoh et al., 1989). The detector comprises a 20 mm-thick Gd 3 Al 2 Ga 3 O 12 (GAGG) scintillator, relay lens optics and a high-speed CMOS camera (Hamamatsu, C11440-22CU). This detector resolved a 1 mm line-and-space pattern.
A Ge slab of dimensions 10 mm (width) Â 14 mm (height) Â 0.5 mm (thickness) and the (001) surface was prepared for the super-Borrmann topography experiment. The slab was rotated in the clockwise direction around the [100] axis until bright spots corresponding to the reflections from two adjacent {111} planes, for example (111) and 1 111 À Á , could be recognized on a fluorescent sheet. It is clear that this multiple (n-beam) diffraction from a single crystal is not considered to be so-called umweganregung (Reninger, 1937a,b) but simply simultaneous excitation of the plural diffractions. After confirming the double fluorescent spots by the two 111 reflections on the sheet, the images formed by the forward-transmitted beam were directly captured by the CMOS camera. As demonstrated in Fig. 3, an adjacent pair of {111} planes was selected by rotating the slab 90 clockwise around the normal to the (001) slab surface.
Figs. 4(a)-4(c) show a fluorescent spot from (a) the directly transmitted X-ray beam denoted as '0', (b) the direct beam and the 111 reflected beam, and (c) the direct beam, the 111 reflected beam and the 1 111 reflected beam. It can be easily noticed that the triple fluorescent spots in Fig. 4(c) are much brighter than those in Fig. 4(b), indicating the super-Borrmann effect. The shining light on the right-hand side of Fig. 4(c) is due to a specular reflection by the Ge crystal surface from the 111 reflection spot on the fluorescent sheet. After the triple fluorescent spots were recognized with nearly the same brightness by sample rotation adjustment around [100] and [001], the topographic image formed by the transmitted beam was captured by the CMOS camera. During the usual Borrmann topography adjustment procedure, no clear dislocation images were recognized on the monitor.    Considering the X-ray source is approximately 1.2 mm Â 1.2 mm in size, four shots of topographic images are pasted together to achieve a wide-area topograph. Regarding the usual Borrmann topography (two-beam case), the images of the dislocations correspond to local lower transmitted intensities (in the forward-refracted direction) or lower diffraction intensities (along the diffraction direction), since crystal imperfection can destroy the Borrmann effect. This is also true for the current super-Borrmann topography (three-beam case), since double excitation of the 111 and 1 111 reflections only enhances the Borrmann effect, i.e. black contrast on the camera monitor corresponds to the local lower diffraction intensity and white contrast corresponds to the local higher intensity, a phenomenon contradictory to that on negative film.
There are four combinations of two adjacent 111 reflections, i.e. 111 and 1 111 reflections (called A-type), 1 111 and 1 1 1 11 reflections (B-type), 1 1 1 11 and 1 1 11 reflections (C-type), and 1 1 11 and 111 reflections (D-type). Additionally, there are two combinations of diagonal 111 reflections, i.e. 111 and 1 1 1 11 reflections (E-type), and 1 111 and 1 1 11 reflections (F-type). Therefore, if one observes dislocation images disappearing only in an A-type topograph, the dislocation should have a Burgers vector of ð1=2Þ 01 1 1 Â Ã , considering this vector is commonly perpendicular to both [111] and 1 111 Â Ã . Similarly, from the invisibility rule g Á b = 0, where g is the diffraction vector and b the dislocation Burgers vector, B-, C-and D-type topographs do not include any images of the dislocations with Burgers vectors of, respectively, (1=2Þ 101 ½ , ð1=2Þ 011 ½ and ð1=2Þ 1 101 Â Ã . However, the combination of diagonal 111 reflections (E-and F-type) does not develop into the super-Borrmann effect owing to the existence of 220 instead of 200 in equation (2). According to the partial lack of the super-Borrmann conditions, Burgers vectors of all the dislocations cannot be determined completely by only observing the A-, B-, C-and D-type topographs. Nevertheless, we can conclude that the Burgers vector of the dislocations disappearing only on the A-type topograph should belong to ð1=2Þ 01 1 1 Â Ã . For example, some parts of A-and B-type topographs are shown in Figs. 6(a) and 6(b), respectively, for the same part of the specimen. The dislocation configurations circled in red can be seen in both topographs.

Conclusions
In this study, we conducted synchrotron X-ray topography exerting the super-Borrmann effect for imaging dislocations using a CMOS camera. Forward-transmitted X-rays can reveal dislocations in relatively thick crystals by simultaneously exciting a pair of adjacent {111} planes owing to the super-Borrmann effect. Super-Borrmann topographs can be captured for relatively thick crystals, even when a conventional Lang X-ray topography technique is difficult to apply.
Prior to the experiment, the minimum attenuation coefficients A min and A min forand -polarizations, respectively, of the incident X-rays in the three-beam (super-Borrmann) case were calculated. It was found that A min and A min were almost 20 times larger than those in the two-beam (usual Borrmann effect) case.
Although it is possible to determine Burgers vectors for some of the dislocations based on the invisibility criteria, it is difficult to finalize the Burgers vectors of most dislocations considering that the employment of a pair of diagonal {111} planes does not produce the super-Borrmann effect.
In addition to the topographs taken by employing forwardtransmitted X-rays under multiple-diffraction conditions (bright-field X-ray topographs), the forward-transmitted X-rays riding on the super-Borrmann effect also reveal dislocations existing in comparatively thick crystals by simultaneously exciting a pair of adjacent {111} planes such as (111)  and 1 111 À Á . Therefore, this study deals with X-ray topography  X-ray topograph of a germanium slab taken by simultaneous 111 and 1 111 reflections without deformation correction. Four shots of images are pasted together to obtain a wide-area topograph.
using synchrotron radiation performed under three-beam multiple diffraction conditions exerting the super-Borrmann effect for thick Ge crystals. Future research will attempt to experimentally detect dislocation behaviors around the very initial growth stage in the necking parts of dislocation-free silicon crystals.
It was clarified that forward-transmitted X-rays using synchrotron radiation can be used to confirm the efficacy for capturing topographs not only under usual multiple diffraction conditions but also under super-Borrmann conditions.